One of the things we often do with math is have students memorize algorithms–which is fine! Good, even! Efficiency and automaticity matter, especially for students with low to average working memories. That is, most students, unlike people who majored in mathematics and then became math teachers, don’t have the mental “space” to derive the procedure every time.
However, what happens is that I get these students later, in algebra, and they cannot tell me why we Keep-Change-Change or “flip-and-multiply” when dividing fractions. While this might not be a significant impediment in K-7 math, this lack of understanding makes algebra difficult. Therefore, when we hit complex fractions, I take a long time for direct instruction on why we do this.
First, I model a simple division problem.
I ask for volunteers to explain why we do this. Generally speaking, nobody can. (If they could, they probably wouldn’t be in my class.)
Then, we have a discussion about the nature of the division symbol. This is always a fun conversation when I point out that this is actually a symbol for a fraction. Circling back to the original problem, I ask if there is a reason why I cannot do this:
Everyone agrees that stacking fractions is technically allowed. However, it’s unhelpful. So then I remind them of the power of 1.
Finally, I show them how to apply this power.
In this case, we’re flipping and multiplying so that we end up with 1 on the bottom of the fraction. This helps us scoot the information out of the denominator. But, we can’t just go around randomly multiplying, because that would change the nature of the original item. Therefore, we create another 1, by stacking the fraction. Anything divided by itself is 1. As a result, we are only multiplying by 1, in a sneaky way to get rid of our original denominator.
As you can see, this gives us another round of something divided by itself, which is also equal to 1.
Last but not least, anything divided by 1 is simply itself.
When we “keep-change-change” or “flip-and-multiply” we are rightly reducing the cognitive load for our younger students. This is a whole lot going on beneath the surface of dividing fractions, and our 9 and 10 year olds don’t need to be burdened with it as they perfect their automaticity. But my older students, for whom dividing fractions is automatic, can and should handle this knowledge. Generally, I get an entire class of, “Whoa!” when we finish this out.
Now, I extrapolate to variables, in the same slide, making a running comparison as I go. Usually I color code it, so that “d” travels in hot pink as the process happens. This small quirk is incredibly helpful for some students, but I don’t know how to do that in a blog post.
Once I’ve painstakingly demonstrated the process, I ask for a mood check.
1) I can do this by myself, thanks.
2) I can follow along, but it’s still a little fuzzy.
3) What are we doing, again?
Generally, 80% or more of the class is somewhere in 1 or 2. Then, I hit it three or four more times, color coding as we go and gradually releasing the process to the students. By the end of the last example, the class should be able to tell me what to do next.
At this point, I do another mood check–there will always be one or two students who have noped out–and then send them off for independent work, with support from the computer adaptive software.
This lesson is one of my favorites, every year, because I love de-mystifying math for them. Now they know the underlying reasons, and we’ll keep applying this over the next few months.